Compile Once, Differentiate Everywhere: A Differentiable Meta-Circular Interpreter

AI systems increasingly propose executable scientific models whose value depends on both their symbolic structure and their fitted continuous parameters. This makes parameter calibration the bottleneck of program-and-parameter co-search: an outer loop can generate thousands of candidate programs, but each needs an inner gradient-based optimization before it can be assessed. Staging each candidate into its own differentiable graph makes individual models fast but sacrifices the program-as-data property that keeps search fluid; interpreter-based approaches preserve programs as runtime data but pay interpreter overhead that dominates the numerical work. We present the Native Differentiable Virtual Machine (NDVM), a runtime representation that differentiates executable programs without compiling each candidate into a separate graph. NDVM separates symbolic structure from differentiable numeric state: tags, symbols, environments, and control remain native runtime data, while numeric payloads live in dense batched buffers with exact reverse-mode gradients recorded along the realized execution trace, so one evaluator walk is amortized across large populations of parameter vectors. A locked cost model of a real differentiable self-hosted Scheme interpreter motivates the design. We realize NDVM as a native runtime with forward and gradient equivalence to the reference backend, about 60x per-lane batch amortization, near-linear multicore scaling, and two independent front ends. In fixed-budget co-search over LLM-proposed programs, NDVM reaches high-quality solutions about 24x sooner in wall-clock time, suggesting runtime differentiation as a practical systems foundation for scientific discovery workflows.

A central challenge in program induction has long been the trade-off between symbolic and neural approaches. Symbolic methods offer compositional generalisation and data efficiency, yet their scalability is constrained by formalisms such as domain-specific languages (DSLs), which are labour-intensive to create and may not transfer to new domains. In contrast, neural networks flexibly learn from data but tend to generalise poorly in compositional and out-of-distribution settings. We bridge this divide with an instance of a Latent Adaptation Network architecture named Neural Language Interpreter (NLI), which learns its own discrete, symbolic-like programming language end-to-end. NLI autonomously discovers a vocabulary of primitive operations and uses a novel differentiable neural executor to interpret variable-length sequences of these primitives. This allows NLI to represent programs that are not bound to a constant number of computation steps, enabling it to solve more complex problems than those seen during training. To make these discrete, compositional program structures amenable to gradient-based optimisation, we employ the Gumbel-Softmax relaxation, enabling the entire model to be trained end-to-end. Crucially, this same differentiability enables powerful test-time adaptation. At inference, NLI's program inductor provides an initial program guess. This guess is then refined via gradient descent through the neural executor, enabling efficient search for the neural program that best explains the given data. We demonstrate that NLI outperforms in-context learning, test-time training, and continuous latent program networks on tasks that require combinatorial generalisation and rapid adaptation to unseen tasks. Our results establish a new path toward models that combine the compositionality of discrete languages with the gradient-based search and end-to-end learning of neural networks.

Scientific machine learning often requires combining known physics with unknown parameters or correction terms learned from data. Existing approaches either ignore known structure, encode it as a soft penalty, or require hand-written PyTorch code for each equation. We present The Neural Compiler, a system that translates programs written in a first-order Scheme-like expression language into frozen, differentiable PyTorch modules. These modules match the source program to floating-point precision and provide gradients through autograd. In hybrid models, the compiled module encodes known physics exactly while learned components model the unknown remainder. We evaluate the compiler across six experiment domains: Feynman physics equations, Lotka-Volterra dynamics, a damped pendulum, a one-dimensional heat equation, three-dimensional vector mechanics, and compositional generalization. Compiled modules match hand-coded PyTorch implementations numerically for single equations, showing no accuracy loss from compilation. With only 1 to 4 trainable parameters, compiled models recover physical constants to less than 1 percent error in most cases, while standard PINN baselines with more than 8500 parameters show 7 to 93 percent error. Compiled modules also compose with zero error, while neural approximations can accumulate large errors in deep composition chains. The main value of the compiler is not improved accuracy over hand-coded equations, but systematic composability: it generates correct, differentiable modules from symbolic specifications without rewriting each equation by hand. The system supports 51 primitive operations, including vector and matrix algebra, enabling PDE discretizations and hybrid scientific models. This string-in, module-out interface also provides a natural target for large language models that translate scientific descriptions into executable differentiable modules.